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Interesting puzzle that is quite hard to stop.

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4 moves, as far as I can tell.

[FEN ""]
1. e3 e5 2. Ke2 Qh4 3. Kf3 d6 4. Qe2 e4#

Another one:

[FEN ""]
1. d3 d5 2. Kd2 e5 3. Kc3 Be6 4. Qd2 d4#

And one more:

[FEN ""]
1. e3 e5 2. Ke2 d5 3. Kd3 Qf6 4. Qe2 e4#

Same theme for all of them, really.

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Mate in 3 moves seems impossible. – Dag Oskar Madsen Jan 4 at 19:15
    
Indeed; also it seems that the shortest mates must take place with white king positioned at third row. Could be interesting to find shortest checkmate sequences for white king positioned at different (1-5) rows. – GloriaVictis Jan 4 at 19:22
    
And two extra versions of each of these, by placing a knight or bishop on d2/e2 instead of the queen. – David Richerby Jan 4 at 20:30
    
@DavidRicherby Indeed, there's even more. For example placing queen on g5 instead of h4 (and pawn on d5/f5 instead of d6) in the first diagram, as well as using a5/c5 pawn or d6 queen instead of f8 bishop to cover b4 square in second diagram (pawn on e5 is not crucial, since queen covers the d4 pawn). Seemed unnecessary to include, since the overarching theme remains unchanged. – GloriaVictis Jan 4 at 20:47
    
@GloriaVictis Ah, I'd not noticed there were so many alternatives. That being the case, I agree there's not much point listing them all. – David Richerby Jan 4 at 21:09

I think it is 5 moves.

[FEN ""]

1. a3 e5 
2. a4 e4
3. a5 e3
4. a6 Bc5
5. axb7 exf2#

You can't do it any shorter as you need at least 4 moves to get your pawn to the 2nd line where it can checkmate the white's king and one more move to bring a supporting piece (queen or bishop), so the white king wouldn't be able to capture your pawn.

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3  
-1; while it is true, that for the trivial mate presented in this answer 5 moves is in fact minimum, it is possible to construct a game with fewer moves required. – GloriaVictis Jan 4 at 18:42

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